The Euler–Lagrange equations for several functions of one variable are given here. The notation there is different from that used in this question, so here's an attempt at a translation. On the Wikipedia page, the independent variable is , in the problem here it is . On the Wikipedia page, the dependent variables are , with derivatives . In this problem, there are two dependent variables, and , with having derivative (but I'll call it since the independent variable is ), and the derivative of does not appear. Finally, the Wikipedia page calls the integrand , but here it is .

According to Wikipedia, the E–L equations are

, for .

Translating those into the notation of this problem, I get the two E–L equations to be

(The second one is particularly simple because all the derivatives with respect to are 0.

If those are correct, the you have two simultaneous linear differential equations, which you can represent as

and you solve them by diagonalising the matrix.